Word Finder - velocity

definition

A vector quantity that denotes the rate of change of position with respect to time, or a speed with the directional component.

definition

Rapidity of motion.

definition

The rate of occurrence.

definition

The number of times that an average unit of currency is spent during a specific period of time.

The wind velocity did not exceed 20 km.

The concentration is known, and the conductivity can be measured experimentally; thus the average velocity with which the ions move past each other under the existent electromotive force can be estimated.

When he considered all days irrespective of wind velocity, Mazelle found the influence of temperature obliterated.

After Professor Amund Helfand had, in July 1875, discovered the amazingly great velocity, up to 644 ft.

The velocity of light (q.v.) has been measured with all the precision necessary for the purpose.

Hence the elementary arc divided by the element of time is the rate of change of velocity of the moving-point, or in other words, the velocity in the hodograph is the acceleration in the orbit.

Rising in the high tablelands or on the slopes of the Drakensberg or Lebombo mountains the rivers in their upper courses have a great slope and a high velocity.

In this way the medium velocity of the current may be diminished, and consequently the quantity of water discharged in a given time must, from the effects of friction, be considerably less than that which is computed from theory.

From a collection of the best experiments by previous workers he selected eighty-two (fifty-one on the velocity of water in conduit pipes, and thirty-one on its velocity in open canals); and, discussing these on physical and mechanical principles, he succeeded in drawing up general formulae, which afforded a simple expression for the velocity of running water.

In particular, for a jet issuing into the atmosphere, where p=P, q 2 /2g = h - z, (9) or the velocity of the jet is due to the head k-z of the still free surface above the orifice; this is Torricelli's theorem (1643), the foundation of the science of hydrodynamics.

At the main base in Adelie Land autumn sledging proved impossible, and throughout the winter there was a continuous succession of terrific blizzards, wind with an average velocity of 50 m.p.h.

Simspon concluded that for a given wind velocity dissipation is practically a linear function of ionization.

The other forms of velocity anemometer may be described as belonging to the windmill type.

The second method is in principle extremely simple, consisting merely in multiplying the observed velocity of light by the time which it takes light to travel from the sun to the earth.

The velocity is now well determined; the difficulty is to determine the time of passage.

He supposed that the filaments of water which graze along the sides of the pipe lose a portion of their velocity; that the contiguous filaments, having on this account a greater velocity, rub upon the former, and suffer a diminution of their celerity; and that the other filaments are affected with similar retardations proportional to their distance from the axis of the pipe.

In the Eulerian method the attention is fixed on a particular point of space, and the change is observed there of pressure, density and velocity, which takes place during the motion; but in the Lagrangian method we follow up a particle of fluid and observe how it changes.

I n a straight uniform current of fluid of density p, flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle 0 with the velocity, is oAq cos 0, the product of the density p, the area A, and q cos 0 the component velocity normal to the plane.

Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and 0 the angle which the outward-drawn normal makes with the velocity q at that point, dM/dt = rate of increase of fluid inside the surface, (I) =flux across the surface into the interior _ - f f pq cos OdS, the integral equation of continuity.

A small sphere of the fluid, if frozen suddenly, would retain this angular velocity.

Calling the sum of the pressure and potential head the statical head, surfaces of constant statical and dynamical head intersect in lines on H, and the three surfaces touch where the velocity is stationary.

Thus if d,/ is the increase of 4, due to a displacement from P to P', and k is the component of velocity normal to PP', the flow across PP' is d4 = k.PP'; and taking PP' parallel to Ox, d,, = vdx; and similarly d/ ' = -udy with PP' parallel to Oy; and generally d4,/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock.

The curves 0 = constant and 4, = constant form an orthogonal system; and the interchange of 0 and 4, will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.

For instance, in a uniplanar flow, radially inward towards 0, the flow across any circle of radius r being the same and denoted by 27rm, the velocity must be mfr, and 0=m log r,, y=m0, +4,i =m log re ie, w=m log z.

A single vortex will remain at rest, and cause a velocity at any point inversely as the distance from the axis and perpendicular to its direction; analogous to the magnetic field of a straight electric current.

If other vortices are present, any one may be supposed to move with the velocity due to the others, the resultant stream function being = gy m log r =log IIrm; (9) the path of a vortex is obtained by equating the value of 1P at the vortex to a constant, omitting the rm of the vortex itself.

Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.-A stream-function 4, must be determined to satisfy the conditions v24 =o, throughout the liquid; (I) I =constant, over any fixed boundary; (2) d,t/ds = normal velocity reversed over a solid boundary, (3) so that, if the solid is moving with velocity U in the direction Ox, d4y1ds=-Udy/ds, or 0 +Uy =constant over the moving cylinder; and 4,+Uy=41' is the stream function of the relative motion of the liquid past the cylinder, and similarly 4,-Vx for the component velocity V along Oy; and generally 1,1'= +Uy -Vx (4) is the relative stream-function, constant over a solid boundary moving with components U and V of velocity.

If the liquid is stirred up by the rotation R of a cylindrical body, d4lds = normal velocity reversed dy = - Rx- Ry ds (5) ds 4' + 2 R (x2 + y2) = Y, (6) a constant over the boundary; and 4,' is the current-function of the relative motion past the cylinder, but now V 2 4,'+2R =o, (7) throughout the liquid.

Over a concentric cylinder, external or internal, of radius r=b, 4,'=4,+ Uly =[U I - + Ui]y, (4) and 4" is zero if U 1 /U = (a 2 - b2)/b 2; (5) so that the cylinder may swim for an instant in the liquid without distortion, with this velocity Ui; and w in (I) will give the liquid motion in the interspace between the fixed cylinder r =a and the concentric cylinder r=b, moving with velocity U1.

If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by w = - Uz, we are left with w = Ua2/z, (6) =U(a 2 /r) cos 0= Ua2x/(x2+y2), (7) 4, = -U(a 2 /r) sin 0= -Ua2y/( x2+y2), (8) giving the motion due to the passage of the cylinder r=a with velocity U through the origin 0 in the direction Ox.

If the direction of motion makes an angle 0' with Ox, tan B' = d0 !dam _ ?xy 2 = tan 20, 0 =-10', (9) dy/ y and the velocity is Ua2/r2.

When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (U 2 + V2)b2a a2U+Vi +b a b a2 U z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (U b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?

With v=o, the angular velocity of the cylinder is 2w; in this way the velocity may be calculated of the propagation of ripples and waves on the surface of a vertical whirlpool in a sink.

Another explanation may be given of the sidelong force, arising from the velocity of liquid past a cylinder, which is encircled by a vortex.

The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin Olga, and its thrust is 27rpmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r=a with velocity U reversed, and the cylinder is surrounded by a vortex.

The velocity of a liquid particle is thus (a 2 - b 2)/(a 2 +b 2) of what it would be if the liquid was frozen and rotating bodily with the ellipse; and so the effective angular inertia of the liquid is (a 2 -b 2) 2 /(a 2 +b 2) 2 of the solid; and the effective radius of gyration, solid and liquid, is given by k 2 = 4 (a 2 2), and 4 (a 2 For the liquid in the interspace between a and n, m ch 2(0-a) sin 2E 4) 1 4Rc 2 sh 2n sin 2E (a2_ b2)I(a2+ b2) = I/th 2 (na)th 2n; (8) and the effective k 2 of the liquid is reduced to 4c 2 /th 2 (n-a)sh 2n, (9) which becomes 4c 2 /sh 2n = s (a 2 - b 2)/ab, when a =00, and the liquid surrounds the ellipse n to infinity.

An angular velocity R, which gives components - Ry, Ix of velocity to a body, can be resolved into two shearing velocities, -R parallel to Ox, and R parallel to Oy; and then ik is resolved into 4'1+1'2, such that 4/ 1 -R-Rx 2 and 1//2+IRy2 is constant over the boundary.

In a similar way the more general state of motion may be analysed, given by w =r ch2('-y), y =a+, i, (26) as giving a homogeneous strain velocity to the confocal system; to which may be added a circulation, represented by an additional term in w.

Motion symmetrical about an Axis.-When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function ' can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2s-4 -11'o); and, as before, if d is the increase in due to a displacement of P to P', then k the component of velocity normal to the surface swept out by PP' is such that 274=2.7ryk.PP'; and taking PP' parallel to Oy and Ox, u= -d/ydy, v=dl,t'/ydx, (I) and 1P is called after the inventor, " Stokes's stream or current function," as it is constant along a stream line (Trans.

The vortex advances with a certain velocity; and if an equal circular vortex is generated coaxially with the first, the mutual influence can be observed.

The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u', v', w' the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have u =U +u' - yR +zQ, v =V +v -zP +xR, w=W +w -xQ +yP.

Thus, for example, with = 4Uy 2 (r 2 a 2 -I), r2 = x2 +y 2, (13) for the space inside the sphere r=a, compared with the value of, i' in § 34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface.

Hill's spherical vortex, advancing through the surrounding liquid with uniform velocity.

As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the ellipsoid l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about the centre with components of angular velocity, 7 7, f'; then u= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of angular velocity S21, 522, S23 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.

To determine the motion of a jet which issues from a vessel with plane walls, the vector I must be constructed so as to have a constant (to) (II) the liquid (15) 2, integrals;, (29) (30) (I) direction 0 along a plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant.

The stream lines xBAJ, xA'J' are given by = 0, m; so that if c denotes the ultimate breadth JJ' of the jet, where the velocity may be supposed uniform and equal to the skin velocity Q, m=Qc, c=m/Q.

Ja - u  ?I a -a b -u' sh nS2=sh log (Q)=?a - b a - a' b - u' At x where = co, u = o, and q= go, (O n b - a ' a + a -b a' cio) - ?a-a'?b a-a' q In crossing to the line of flow x'A'P'J', b changes from o to m, so that with q = Q across JJ', while across xx the velocity is qo, so that i n = go.

The motion of a jet impinging on an infinite barrier is obtained by putting j = a, j' = a'; duplicated on the other side of the barrier, the motion reversed will represent the direct collision of two jets of unequal breadth and equal velocity.

The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity U and 15 1 of equation (II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular area a l.

When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity U through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal ellipsoid defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.